How do I desolder these specific points?

You need a bigger soldering iron, as in "more power."

The square connections are in the ground plane of the board. That is the large yellow area they are embedded in. That is a large piece of copper, and there are probably also large copper surfaces on the internal layers of the board.

Copper conducts heat very well, and it also radiates it away.

The large copper areas are basically sucking up all the heat your iron can provide and radiating it away fast enough that it can't get hot enough to melt solder.

The solution is an iron that can put in heat faster than the board can dissipate it.

So, you need an iron with more power.

Many irons are only around 30 watts. You'll need much more than that.

When I've had to do that kind of thing, I borrowed a huge 150 watt iron from my father in law. It isn't intended for electronics, but it has the raw power needed for large copper surfaces.

As for technique, high wattage irons often have wide tips.

I apply some extra solder to the heavy joint with the iron heating just the ground connection.

When that finally melts, I rotate the tip of the iron to heat both pads for that part.

The solder melts pretty quickly, then I can pull the part out. Afterwards (if you need to to replace the part) you can clean the holes with a solder sucker or solder wick.

While you are removing the part, you actually want as much solder as possible on the connection. Removing solder makes it harder to get the part out, not easier.

Is there a special technique or tool I need to perform this task?

Nothing special but try to get hold of a rework station with blower. As pointed out by others, the square ones are indeed ground plane which require a bit more time to heat up due to the large surface area. Using the blower adjust the blow speed and temperature (such that the board doesn't burn up). Apply flux, heat the desired location and with the help of tweezers plug out the caps.

I have also experienced the same problem before, so I plan on dividing my ground plane into smaller sections during routing.

Edit: In Eagle, there is an option of enabling thermals for a pad(enabled by default). Enabling this keeps a bit of gap between the pad and the surrounding plane. This helps in quickly heating the pad before the heat dissipates.

Most likely both pins are connected to planes. The ground plane on bottom layer is obvious, but there is probably a power plane on the other pin as well. So the solder will melt half way through the board (and visually, from your side, it looks melted) however there will still be solid solder inside.

The solution is:

  • Clamp the board to something solid.
  • Have someone grab the cap from the other side with a pair of pliers and pull GENTLY.
  • Apply one >90W soldering iron with flat wide tip to each pin, adding leaded solder to lower the melting point. Yes you need two irons and two hands.

Another method is:

  • Cut a piece of thick (like 4mm2) copper wire long enough to span both of the cap's pin.
  • Place it across the cap's pins.
  • Heat it with a powerful iron (like an instant-on 100W soldering pistol).
  • The copper will spread the heat to both pins. Apply a generous helping of leaded solder to lower the melting point.
  • This method only requires one iron (albeit a beefy one) and two hands, so it is more practical than the previous one.
  • A cut piece of copper wire bent and shaped properly also works wonders to heat all the pins on big SMD chips, connectors, etc, and makes desoldering a lot easier. With a bit of practice, desoldering QFPs without damaging the board is rather simple.

This are the simplest "ghetto" method. If you have a hot air rework station, you can preheat the board, which will make your job a lot easier. Don't use something like a 2000W hot air gun, this is very effective at burning and delaminating boards (after all it is intended to strip paint...)